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Calculus, MET1 2022 VCAA 1b

Find and simplify the rule of `f^{\prime}(x)`, where `f:R \rightarrow R, f(x)=\frac{\cos (x)}{e^x}`.   (2 marks)

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`\frac{-(\sin x+\cos x)}{e^x}`

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 Using the quotient rule

`f(x)` `=\frac{\cos (x)}{e^x}`  
`f^{\prime}(x)` `=\frac{-e^x \sin x-e^x \cos x}{e^{2 x}}`  
  `= \frac{-e^x(\sin x+\cos x)}{e^{2 x}}`  
  `=\frac{-(\sin x+\cos x)}{e^x}`  

Calculus, MET1 2006 ADV 2aii

Differentiate with respect to `x`:

Let  `y=sin x/(x + 1)`.  Find  `dy/dx `.   (2 marks)

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`dy/dx = {cos x (x + 1) – sin x} / (x + 1)^2`

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`y = sinx/(x + 1)`

`d/dx (u/v) = (u^{\prime} v – uv^{\prime})/v^2`

`u` `= sin x` `v` `= x + 1`
`u^{\prime}` `= cos x` `\ \ \ v^{\prime}` `= 1`

 

`:.dy/dx = {cos x (x + 1) – sin x} / (x + 1)^2`

Calculus, MET1 2019 VCAA 1b

Let  `g: R text(\ {−1}) -> R,\ \ g(x) = (sin(pi x))/(x + 1)`.

Evaluate  `g prime(1)`.  (2 marks)

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`-pi/2`

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  `u = sin(pi x)` `v = x + 1`
  `u prime = pi cos(pi x)` `v prime = 1`

 

`g prime(x)` `= (v u prime – u v prime)/v^2`
  `= ((x + 1) ⋅ pi cos(pi x) – sin (pi x))/(x + 1)^2`
`g prime(1)` `= (2 pi cos(pi) – sin(pi))/2^2`
  `= (2 pi(-1) – 0)/4`
  `= -pi/2`

Calculus, MET1 2018 VCAA 1b

Let  `f(x) = (e^x)/(cos(x))`.

Evaluate  `f′(pi)`.  (2 marks)

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`text(See Worked Solutions)`

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`f′(x) = (e^x)/(cos(x))`

`u` `= e^x` `v` `= cos(x)`
`u′` `= e^x` `v′` `= −sin(x)`
`f′(x)` `= (u′v – uv′)/(v^2)`
  `= (e^x · cos(x) + e^x sin(x))/(cos^2(x))`

 

`f′(pi)` `= (e^pi · cospi + e^pi sinpi)/(cos^2 pi)`
  `= (e^pi(−1) + e^pi · 0)/((−1)^2)`
  `= −e^pi`

Calculus, MET1 2012 VCAA 1b

If  `f(x) = x/(sin(x))`,  find  `f prime (pi/2).`  (2 marks)

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`1`

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`text(Using Quotient Rule:)`

`(h/g)′` `= (h′ g – h g′)/g^2`
`f prime (x)` `= (1 xx sin (x) – x cos (x))/(sin x)^2`
`:. f prime (pi/2)` `= (sin (pi/2) – pi/2 xx cos (pi/2))/(sin(pi/2))^2`
  `= (1 – 0)/1^2`
  `= 1`

Calculus, MET1 2009 VCAA 1b

For  `f(x) = (cos(x))/(2x + 2)`  find  `f prime (pi).`  (3 marks)

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`1/(2 (pi + 1)^2)`

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`text(Using Quotient Rule:)`

MARKER’S COMMENT: A majority of students did not substitute  `x=pi`  correctly.
`(g/h)′` `= (g′ h – gh′)/h^2`
`f prime (x)` `= (-sin (x) (2x + 2) – 2 cos (x))/(2x + 2)^2`
`:. f prime (pi)` `= (-sin (pi) (2pi + 2) – 2 cos (pi))/(2pi + 2)^2`
  `= (0 – 2 (-1))/[2 (pi + 1)]^2`
  `= 2/(4(pi + 1)^2)`
  `= 1/(2 (pi + 1)^2)`

Calculus, MET1 2016 VCAA 1a

Let `y = (cos(x))/(x^2 + 2)`.

Find  `(dy)/(dx)`.  (2 marks)

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`(−x^2sin(x) – 2sin(x) – 2xcos(x))/((x^2 + 2)^2)`

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`text(Using Quotient Rule:)`

`(h/g)′` `= (h′g – hg′)/(g^2)`
`(dy)/(dx)` `= (−sin(x)(x^2 + 2) – cos(x)(2x))/((x^2 + 2)^2)`
  `= (−x^2sin(x) – 2sin(x) – 2xcos(x))/((x^2 + 2)^2)`